I. To determine the maximum tolerated dose of the combination of veliparib (ABT-888) and weekly topotecan (topotecan hydrochloride) in adult patients with advanced solid tumors. (Phase I) II. To identify any anti-tumor activity of this treatment combination, as assessed by objective response in patients with advanced solid tumors. (Phase I) III. To assess the confirmed response rate for patients with epithelial ovarian cancer, fallopian tube cancer or primary peritoneal carcinoma treated with the combination of ABT-888 and weekly topotecan.